Introduction to connected operators
Henk J.A.M. Heijmans · Centrum Wiskunde & Informatica (CWI), the national research institute for mathematics and computer science in the Netherlands · 1999
indicator function.If S is a statement then [S] denotes the Boolean value (0 or 1) indicating whether S is true or false.Thus we can write [h e X] instead of X (h).For other unknown notation and terminology, the reader may refer to Chapter 5. CONNECTIVITY AND RECONSTRUCTIONBy C we denote the subcollection of all subsets of 71} which are 8-connected.The family C is a typical example of a connectivity class; see also Section 6.7.Figure 6-4.A connected operator applied to the left image can result in the image at the right but not in the one in the middle.A connected operator acts on the grains of the foreground and background in an all-or-nothing way: either the grain is left untouched or deleted altogether.This means in particular that the borders in the image cannot be broken or changed, but only deleted.An illustration is given in Fig. 6-4: the middle image cannot be the output of a connected operator applied to the image at the left.However, the right image may result from a connected operator.PROPOSITION 6-3.An operator 1/; is connected if and only if X 6.1/;(X) consists of grains of X and xc,Jor every X s; '71}.PROOF."only if": assume that 1/; is connected; then P(l/;(X)) is coarser than P(X).We must prove that for every h E X!:::..1/;(X), the entire part P(X, h) lies in X 6.1/; (X).We have to consider two cases: h E X and h fj.X. h EX: thus h rj 1/;(X).Then P(X, h) s; P(l/;(X), h) leads to Yh(X) s; Yh(l/;(X)c).But this means that Yh(X) s; X!:::,,1/;(X).h fj.X: then h E 1/;(X), and P(X, h) s;Again, we must distinguish between the cases h E X and h f.X.We consider only the first case; the second is treated analogously.We must show that Yh(X) s:; P(l/;(X), h).Suppose Yh(X) i 1/;(X); then there is a point k such that k E Yh (X) and k fj.1/; (X).Now k E X 6.1/; (X), which yields that Yk(X) £:; X!:::..1/;(X).However, Yk(X) = Yh(X), whence we conclude that Yh(X) s:; 1/;(X)c, and thus Yh(X) £:; Yh(l/;(X)C) = P(l/;(X), h).• In fact, the condition that X !:::,, 1/;(X) consists of grains of X and xc, consists of two parts, namely that X \ 1/;(X) consists of grains of X, and that 1/;(X) \ X consists of grains of xc.